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We define game semantics for the constructive $\mu$-calculus and prove its equivalence to bi-relational semantics. As an application, we use the game semantics to prove that the $\mu$-calculus collapses to modal logic over the modal logic $\mathsf{IS5}$. We then show the completeness of $\mathsf{IS5}$ extended with fixed-point operators.

We discuss the second-order differential uniformity of vectorial Boolean functions. The closely related notion of second-order zero differential uniformity has recently been studied in connection to resistance to the boomerang attack. We prove that monomial functions with univariate form $x^d$ where $d=2^{2k}+2^k+1$ and $\gcd(k,n)=1$ have optimal second-order differential uniformity. Computational results suggest that, up to affine equivalence, these might be the only optimal cubic power functions. We begin work towards generalising such conditions to all monomial functions of algebraic degree 3. We also discuss further questions arising from computational results.

In this work, we will study a numerical method that allows finding an approximation of the exact solution for a in-situ combustion model using the nonlinear mixed complementary method, which is a variation of the Newtons method for solving nonlinear systems based on an implicit finite difference scheme and a nonlinear algorithm mixed complementarity, FDA-MNCP. The method has the advantage of provide a global convergence in relation to the finite difference method and method of Newton that only has local convergence. The theory is applied to model in-situ combustion, which can be rewritten in the form of mixed complementarity also we do a comparison with the FDA-NCP method

We derive a robust error estimate for a recently proposed numerical method for $\alpha$-dissipative solutions of the Hunter-Saxton equation, where $\alpha \in [0, 1]$. In particular, if the following two conditions hold: i) there exist a constant $C > 0$ and $\beta \in (0, 1]$ such that the initial spatial derivative $\bar{u}_{x}$ satisfies $\|\bar{u}_x(\cdot + h) - \bar{u}_x(\cdot)\|_2 \leq Ch^{\beta}$ for all $h \in (0, 2]$, and ii), the singular continuous part of the initial energy measure is zero, then the numerical wave profile converges with order $O(\Delta x^{\frac{\beta}{8}})$ in $L^{\infty}(\mathbb{R})$. Moreover, if $\alpha=0$, then the rate improves to $O(\Delta x^{\frac{1}{4}})$ without the above assumptions, and we also obtain a convergence rate for the associated energy measure - it converges with order $O(\Delta x^{\frac{1}{2}})$ in the bounded Lipschitz metric. These convergence rates are illustrated by several examples.

The preservation of stochastic orders by distortion functions has become a topic of increasing interest in the reliability analysis of coherent systems. The reason of this interest is that the reliability function of a coherent system with identically distributed components can be represented as a distortion function of the common reliability function of the components. In this framework, we study the preservation of the excess wealth order, the total time on test transform order, the decreasing mean residual live order, and the quantile mean inactivity time order by distortion functions. The results are applied to study the preservation of these stochastic orders under the formation of coherent systems with exchangeable components.

In this paper, we introduce and analyze a numerical scheme for solving the Cauchy-Dirichlet problem associated with fractional nonlinear diffusion equations. These equations generalize the porous medium equation and the fast diffusion equation by incorporating a fractional diffusion term. We provide a rigorous analysis showing that the discretization preserves main properties of the continuous equations, including algebraic decay in the fractional porous medium case and the extinction phenomenon in the fractional fast diffusion case. The study is supported by extensive numerical simulations. In addition, we propose a novel method for accurately computing the extinction time for the fractional fast diffusion equation and illustrate numerically the convergence of rescaled solutions towards asymptotic profiles near the extinction time.

The main purpose of this paper is to design a local discontinuous Galerkin (LDG) method for the Benjamin-Ono equation. We analyze the stability and error estimates for the semi-discrete LDG scheme. We prove that the scheme is $L^2$-stable and it converges at a rate $\mathcal{O}(h^{k+1/2})$ for general nonlinear flux. Furthermore, we develop a fully discrete LDG scheme using the four-stage fourth order Runge-Kutta method and ensure the devised scheme is strongly stable in case of linear flux using two-step and three-step stability approach under an appropriate time step constraint. Numerical examples are provided to validate the efficiency and accuracy of the method.

In this study, we address the challenge of solving elliptic equations with quasiperiodic coefficients. To achieve accurate and efficient computation, we introduce the projection method, which enables the embedding of quasiperiodic systems into higher-dimensional periodic systems. To enhance the computational efficiency, we propose a compressed storage strategy for the stiffness matrix by its multi-level block circulant structure, significantly reducing memory requirements. Furthermore, we design a diagonal preconditioner to efficiently solve the resulting high-dimensional linear system by reducing the condition number of the stiffness matrix. These techniques collectively contribute to the computational effectiveness of our proposed approach. Convergence analysis shows the spectral accuracy of the proposed method. We demonstrate the effectiveness and accuracy of our approach through a series of numerical examples. Moreover, we apply our method to achieve a highly accurate computation of the homogenized coefficients for a quasiperiodic multiscale elliptic equation.

We show that the parameters of a $k$-mixture of inverse Gaussian or gamma distributions are algebraically identifiable from the first $3k-1$ moments, and rationally identifiable from the first $3k+2$ moments. Our proofs are based on Terracini's classification of defective surfaces, careful analysis of the intersection theory of moment varieties, and a recent result on sufficient conditions for rational identifiability of secant varieties by Massarenti--Mella.

We consider non-linear Bayesian inverse problems of determining the parameter $f$. For the posterior distribution with a class of Gaussian process priors, we study the statistical performance of variational Bayesian inference to the posterior with variational sets consisting of Gaussian measures or a mean-field family. We propose certain conditions on the forward map $\mathcal{G}$, the variational set $\mathcal{Q}$ and the prior such that, as the number $N$ of measurements increases, the resulting variational posterior distributions contract to the ground truth $f_0$ generating the data, and derive a convergence rate with polynomial order or logarithmic order. As specific examples, we consider a collection of non-linear inverse problems, including the Darcy flow problem, the inverse potential problem for a subdiffusion equation, and the inverse medium scattering problem. Besides, we show that our convergence rates are minimax optimal for these inverse problems.